This paper presents a new stochastic point process able to analyse the failure pattern of complex mechanical systems experiencing both early failures and degradation phenomena, and operating so long that the intensity function approaches a finite asymptote as the system age grows. We illustrate the characteristics of the proposed model. Then we give the maximum likelihood estimators of the model parameters, and some quantities of interest such as the s-expected number of early failures, the time of minimum intensity, and the minimum intensity value. Inference on the number of failures that will occur in future time intervals is also provided. We applied the proposed model to real failure data from the powertrain system of two buses operating in urban routes in the city of Naples.